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Matrix Example

fricas
m1 := matrix([ [1, x/y, 0], [0, 1, 0], [0, 0, 1] ])

\label{eq1}\left[ 
\begin{array}{ccc}
1 &{x \over y}& 0 
\
0 & 1 & 0 
\
0 & 0 & 1 
(1)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
m2 := matrix([ [1, 0, 0], [0, 1, y/x], [0, 0, 1] ])

\label{eq2}\left[ 
\begin{array}{ccc}
1 & 0 & 0 
\
0 & 1 &{y \over x}
\
0 & 0 & 1 
(2)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
m3 := matrix([ [-1/y, 0, 0], [0, y/(x*z), 0], [0, 0, x/(x*y-z*y)] ])

\label{eq3}\left[ 
\begin{array}{ccc}
-{1 \over y}& 0 & 0 
\
0 &{y \over{x \  z}}& 0 
\
0 & 0 & -{x \over{{y \  z}-{x \  y}}}
(3)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
m4 := matrix([ [1, 0, 0], [0, 1, 0], [0, y/x, 1] ])

\label{eq4}\left[ 
\begin{array}{ccc}
1 & 0 & 0 
\
0 & 1 & 0 
\
0 &{y \over x}& 1 
(4)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
m5 := matrix([ [1, 0, 0], [z/y, 1, 0], [0, 0, 1] ])

\label{eq5}\left[ 
\begin{array}{ccc}
1 & 0 & 0 
\
{z \over y}& 1 & 0 
\
0 & 0 & 1 
(5)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
m6 := matrix([ [0, 0, 1], [0, 1, 0], [1, 0, 0] ])

\label{eq6}\left[ 
\begin{array}{ccc}
0 & 0 & 1 
\
0 & 1 & 0 
\
1 & 0 & 0 
(6)
Type: Matrix(NonNegativeInteger)
fricas
T := m1*m2*m3*m4*m5*m6

\label{eq7}\left[ 
\begin{array}{ccc}
-{x \over{{y \  z}-{x \  y}}}& -{x \over{{{z}^{2}}-{x \  z}}}& -{z \over{{y \  z}-{x \  y}}}
\
-{1 \over{z - x}}& -{y \over{{{z}^{2}}-{x \  z}}}& -{1 \over{z - x}}
\
-{x \over{{y \  z}-{x \  y}}}& -{1 \over{z - x}}& -{z \over{{y \  z}-{x \  y}}}
(7)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
M := matrix([ [0, -z, y], [z, 0, -x], [-y, x, 0] ])

\label{eq8}\left[ 
\begin{array}{ccc}
0 & - z & y 
\
z & 0 & - x 
\
- y & x & 0 
(8)
Type: Matrix(Polynomial(Integer))
fricas
R := matrix([ [r, 0, 0], [0, r, 0], [0, 0, r] ])

\label{eq9}\left[ 
\begin{array}{ccc}
r & 0 & 0 
\
0 & r & 0 
\
0 & 0 & r 
(9)
Type: Matrix(Polynomial(Integer))
fricas
T - R

\label{eq10}\left[ 
\begin{array}{ccc}
{{-{r \  y \  z}+{r \  x \  y}- x}\over{{y \  z}-{x \  y}}}& -{x \over{{{z}^{2}}-{x \  z}}}& -{z \over{{y \  z}-{x \  y}}}
\
-{1 \over{z - x}}&{{-{r \ {{z}^{2}}}+{r \  x \  z}- y}\over{{{z}^{2}}-{x \  z}}}& -{1 \over{z - x}}
\
-{x \over{{y \  z}-{x \  y}}}& -{1 \over{z - x}}&{{{{\left(-{r \  y}- 1 \right)}\  z}+{r \  x \  y}}\over{{y \  z}-{x \  y}}}
(10)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
determinant(%)

\label{eq11}{{{\left(-{{{r}^{3}}\  y}-{{r}^{2}}\right)}\ {{z}^{2}}}+{{\left({{{r}^{3}}\  x \  y}-{{{r}^{2}}\  x}\right)}\  z}-{{{r}^{2}}\ {{y}^{2}}}- 1}\over{{y \ {{z}^{2}}}-{x \  y \  z}}(11)
Type: Fraction(Polynomial(Integer))
fricas
T * vector([bx, by, bc])

\label{eq12}\begin{array}{@{}l}
\displaystyle
\left[{{-{bc \ {{z}^{2}}}-{bx \  x \  z}-{ \hbox{ by } \  x \  y}}\over{{y \ {{z}^{2}}}-{x \  y \  z}}}, \:{{{{\left(- bx - bc \right)}\  z}-{ \hbox{ by } \  y}}\over{{{z}^{2}}-{x \  z}}}, \: \right.
\
\
\displaystyle
\left.{{-{bc \  z}-{ \hbox{ by } \  y}-{bx \  x}}\over{{y \  z}-{x \  y}}}\right] 
(12)
Type: Vector(Fraction(Polynomial(Integer)))

fricas
R := matrix([ [cos(t), -sin(t), 0], [sin(t), cos(t), 0], [0,0,1] ])

\label{eq13}\left[ 
\begin{array}{ccc}
{\cos \left({t}\right)}& -{\sin \left({t}\right)}& 0 
\
{\sin \left({t}\right)}&{\cos \left({t}\right)}& 0 
\
0 & 0 & 1 
(13)
Type: Matrix(Expression(Integer))
fricas
T := transpose(R)

\label{eq14}\left[ 
\begin{array}{ccc}
{\cos \left({t}\right)}&{\sin \left({t}\right)}& 0 
\
-{\sin \left({t}\right)}&{\cos \left({t}\right)}& 0 
\
0 & 0 & 1 
(14)
Type: Matrix(Expression(Integer))
fricas
e11 := matrix( [ [1,0,0], [0,0,0], [0,0,0] ] )

\label{eq15}\left[ 
\begin{array}{ccc}
1 & 0 & 0 
\
0 & 0 & 0 
\
0 & 0 & 0 
(15)
Type: Matrix(NonNegativeInteger)
fricas
e12 := matrix( [ [0,1,0], [0,0,0], [0,0,0] ] )

\label{eq16}\left[ 
\begin{array}{ccc}
0 & 1 & 0 
\
0 & 0 & 0 
\
0 & 0 & 0 
(16)
Type: Matrix(NonNegativeInteger)
fricas
e13 := matrix( [ [0,0,1], [0,0,0], [0,0,0] ] )

\label{eq17}\left[ 
\begin{array}{ccc}
0 & 0 & 1 
\
0 & 0 & 0 
\
0 & 0 & 0 
(17)
Type: Matrix(NonNegativeInteger)
fricas
e21 := matrix( [ [0,0,0], [1,0,0], [0,0,0] ] )

\label{eq18}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
1 & 0 & 0 
\
0 & 0 & 0 
(18)
Type: Matrix(NonNegativeInteger)
fricas
e22 := matrix( [ [0,0,0], [0,1,0], [0,0,0] ] )

\label{eq19}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 1 & 0 
\
0 & 0 & 0 
(19)
Type: Matrix(NonNegativeInteger)
fricas
e23 := matrix( [ [0,0,0], [0,0,1], [0,0,0] ] )

\label{eq20}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 1 
\
0 & 0 & 0 
(20)
Type: Matrix(NonNegativeInteger)
fricas
e31 := matrix( [ [0,0,0], [0,0,0], [1,0,0] ] )

\label{eq21}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 0 
\
1 & 0 & 0 
(21)
Type: Matrix(NonNegativeInteger)
fricas
e32 := matrix( [ [0,0,0], [0,0,0], [0,1,0] ] )

\label{eq22}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 0 
\
0 & 1 & 0 
(22)
Type: Matrix(NonNegativeInteger)
fricas
e33 := matrix( [ [0,0,0], [0,0,0], [0,0,1] ] )

\label{eq23}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 0 
\
0 & 0 & 1 
(23)
Type: Matrix(NonNegativeInteger)
fricas
T*e11*R

\label{eq24}\left[ 
\begin{array}{ccc}
{{\cos \left({t}\right)}^{2}}& -{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 
\
-{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}&{{\sin \left({t}\right)}^{2}}& 0 
\
0 & 0 & 0 
(24)
Type: Matrix(Expression(Integer))
fricas
T*e12*R

\label{eq25}\left[ 
\begin{array}{ccc}
{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}&{{\cos \left({t}\right)}^{2}}& 0 
\
-{{\sin \left({t}\right)}^{2}}& -{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 
\
0 & 0 & 0 
(25)
Type: Matrix(Expression(Integer))
fricas
T*e13*R

\label{eq26}\left[ 
\begin{array}{ccc}
0 & 0 &{\cos \left({t}\right)}
\
0 & 0 & -{\sin \left({t}\right)}
\
0 & 0 & 0 
(26)
Type: Matrix(Expression(Integer))
fricas
T*e21*R

\label{eq27}\left[ 
\begin{array}{ccc}
{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& -{{\sin \left({t}\right)}^{2}}& 0 
\
{{\cos \left({t}\right)}^{2}}& -{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 
\
0 & 0 & 0 
(27)
Type: Matrix(Expression(Integer))
fricas
T*e22*R

\label{eq28}\left[ 
\begin{array}{ccc}
{{\sin \left({t}\right)}^{2}}&{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 
\
{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}&{{\cos \left({t}\right)}^{2}}& 0 
\
0 & 0 & 0 
(28)
Type: Matrix(Expression(Integer))
fricas
T*e23*R

\label{eq29}\left[ 
\begin{array}{ccc}
0 & 0 &{\sin \left({t}\right)}
\
0 & 0 &{\cos \left({t}\right)}
\
0 & 0 & 0 
(29)
Type: Matrix(Expression(Integer))
fricas
T*e31*R

\label{eq30}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 0 
\
{\cos \left({t}\right)}& -{\sin \left({t}\right)}& 0 
(30)
Type: Matrix(Expression(Integer))
fricas
T*e32*R

\label{eq31}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 0 
\
{\sin \left({t}\right)}&{\cos \left({t}\right)}& 0 
(31)
Type: Matrix(Expression(Integer))
fricas
T*e33*R

\label{eq32}\left[ 
\begin{array}{ccc}
0 & 0 & 0 
\
0 & 0 & 0 
\
0 & 0 & 1 
(32)
Type: Matrix(Expression(Integer))




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